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PROFESSOR: Let's begin.
9
00:00:26,540 --> 00:00:31,190
Today we're going to continue
the discussion on Ito calculus.
10
00:00:31,190 --> 00:00:33,840
I briefly introduced you
to Ito's lemma last time,
11
00:00:33,840 --> 00:00:38,510
but let's begin by
reviewing it and stating it
12
00:00:38,510 --> 00:00:39,910
in a slightly more general form.
13
00:00:42,570 --> 00:00:45,790
Last time what we did was we
did the quadratic variation
14
00:00:45,790 --> 00:00:51,830
of Brownian motion,
Brownian process.
15
00:00:57,140 --> 00:01:01,160
We defined the Brownian
process, Brownian motion,
16
00:01:01,160 --> 00:01:11,700
and then showed that it has
quadratic variation, which
17
00:01:11,700 --> 00:01:22,380
can be written in this form--
dB square is equal to dt.
18
00:01:22,380 --> 00:01:33,350
And then we used that to show
the simple form of Ito's lemma,
19
00:01:33,350 --> 00:01:38,080
which says that if f is a
function on the Brownian
20
00:01:38,080 --> 00:01:48,820
motion, then d of f is
equal to f prime of dB_t
21
00:01:48,820 --> 00:01:56,990
plus f double prime of dt.
22
00:01:56,990 --> 00:02:03,730
This additional term was a
characteristic of Ito calculus.
23
00:02:03,730 --> 00:02:06,405
In classical calculus
we only have this term,
24
00:02:06,405 --> 00:02:08,509
but we have this
additional term.
25
00:02:08,509 --> 00:02:10,389
And if you remember,
this happened exactly
26
00:02:10,389 --> 00:02:13,690
because of this
quadratic variation.
27
00:02:13,690 --> 00:02:17,100
Let's review it, and let's do
it in a slightly more general
28
00:02:17,100 --> 00:02:18,530
form.
29
00:02:18,530 --> 00:02:20,630
As you know, we
have a function f
30
00:02:20,630 --> 00:02:24,690
depending on two
variables, t and x.
31
00:02:24,690 --> 00:02:30,830
Now we're interested
in-- we want to evaluate
32
00:02:30,830 --> 00:02:35,956
our information on the
function f(t, B_t).
33
00:02:38,890 --> 00:02:41,120
The second coordinate,
we're planning
34
00:02:41,120 --> 00:02:43,966
to put in the
Brownian motion there.
35
00:02:43,966 --> 00:02:45,590
Then again, let's do
the same analysis.
36
00:02:45,590 --> 00:02:52,530
Can we describe d of f in terms
of these differentiations?
37
00:02:52,530 --> 00:02:58,610
To do that, deflect this, let
me start from Taylor expansion.
38
00:03:04,460 --> 00:03:16,080
f at a point t plus delta
t, x plus delta x by Taylor
39
00:03:16,080 --> 00:03:31,400
expansion for two variables is
f of t of x plus partial of f
40
00:03:31,400 --> 00:03:39,068
over partial of t at t
comma x of delta t plus...
41
00:03:42,980 --> 00:03:44,510
x.
42
00:03:44,510 --> 00:03:47,651
That's the first-order terms.
43
00:03:47,651 --> 00:03:49,150
Then we have the
second-order terms.
44
00:04:27,560 --> 00:04:31,440
Then the third-order
terms, and so on.
45
00:04:31,440 --> 00:04:34,700
That's just Taylor expansion.
46
00:04:34,700 --> 00:04:37,030
If you look at it,
we have a function f.
47
00:04:37,030 --> 00:04:39,370
We want to look at the
difference of f when we change
48
00:04:39,370 --> 00:04:41,661
the first variable a little
bit and the second variable
49
00:04:41,661 --> 00:04:42,830
a little bit.
50
00:04:42,830 --> 00:04:44,984
We start from f of t of x.
51
00:04:44,984 --> 00:04:47,400
In the first-order terms, you
take the partial derivative,
52
00:04:47,400 --> 00:04:50,350
so take del f over del
t, and then multiply
53
00:04:50,350 --> 00:04:52,200
by the t difference.
54
00:04:52,200 --> 00:04:54,350
Second term, you take
the partial derivative
55
00:04:54,350 --> 00:04:57,420
with respect to the second
variable-- partial f
56
00:04:57,420 --> 00:05:02,470
over partial x-- and
then multiply by del x.
57
00:05:02,470 --> 00:05:05,560
That much is enough
for classical calculus.
58
00:05:05,560 --> 00:05:08,000
But then, as we
have seen before,
59
00:05:08,000 --> 00:05:09,750
we ought to look at
the second-order term.
60
00:05:09,750 --> 00:05:14,030
So let's first write
down what it is.
61
00:05:14,030 --> 00:05:16,370
That's exactly what happened
in Taylor expansion,
62
00:05:16,370 --> 00:05:17,140
if you remember.
63
00:05:17,140 --> 00:05:20,130
If you don't remember,
just believe me.
64
00:05:20,130 --> 00:05:23,856
This 1 over 2 times, take the
second derivatives, partial.
65
00:05:26,400 --> 00:05:29,100
Let's write it in
terms of-- yes?
66
00:05:29,100 --> 00:05:31,806
AUDIENCE: [INAUDIBLE]
67
00:05:34,464 --> 00:05:36,978
PROFESSOR: Oh,
yeah, you're right.
68
00:05:36,978 --> 00:05:37,946
Thank you.
69
00:05:45,700 --> 00:05:46,350
Is it good now?
70
00:05:50,060 --> 00:05:52,150
Let's write it as
dt, all these deltas.
71
00:05:56,592 --> 00:05:57,830
I'll just write like that.
72
00:05:57,830 --> 00:06:00,270
I'll just not
write down t and x.
73
00:06:00,270 --> 00:06:05,980
And what we have is f plus
del f over del t dt plus del
74
00:06:05,980 --> 00:06:10,810
f over del x dx plus
the second-order terms.
75
00:06:37,290 --> 00:06:39,990
The only important terms--
first of all, these terms
76
00:06:39,990 --> 00:06:42,620
are important.
77
00:06:42,620 --> 00:06:44,660
But then, if you want
to use x equals B of
78
00:06:44,660 --> 00:06:49,780
t-- so if you're now
interested in f t comma B of t.
79
00:06:49,780 --> 00:06:55,970
Or more generally, if you're
interested in f t plus dt,
80
00:06:55,970 --> 00:07:03,285
f B_t plus d of B_t, then
these terms are important.
81
00:07:03,285 --> 00:07:07,720
If you subtract f of
t of B_t, what you get
82
00:07:07,720 --> 00:07:11,460
is these two terms.
83
00:07:11,460 --> 00:07:16,916
Del f over del t dt
plus del f over del
84
00:07:16,916 --> 00:07:19,970
x-- I'm just writing
this as a second variable
85
00:07:19,970 --> 00:07:22,035
differentiation-- at dB_t.
86
00:07:25,880 --> 00:07:28,550
And then the second-order terms.
87
00:07:28,550 --> 00:07:32,605
Instead of writing it all down,
dt square is insignificant,
88
00:07:32,605 --> 00:07:37,580
and dt comma-- dt times
dB_t also is insignificant.
89
00:07:37,580 --> 00:07:39,910
But the only thing that
matters will be this one.
90
00:07:39,910 --> 00:07:45,000
This is dB_t square, which
you saw is equal to dt.
91
00:07:48,920 --> 00:07:52,990
From the second-order term,
we'll have this term surviving.
92
00:07:52,990 --> 00:08:01,160
1 over 2 partial f over partial
x second derivative, of dt.
93
00:08:01,160 --> 00:08:04,010
That's it.
94
00:08:04,010 --> 00:08:05,860
If you rearrange
it, what we get is
95
00:08:05,860 --> 00:08:18,259
partial f over partial
t plus 1/2 this plus--
96
00:08:18,259 --> 00:08:19,550
and that's the additional term.
97
00:08:25,150 --> 00:08:28,620
If you ask me why these
terms are not important
98
00:08:28,620 --> 00:08:33,150
and this term is important, I
can't really say it rigorously.
99
00:08:33,150 --> 00:08:36,929
But if you think about dB_t
square equals dt, then d times
100
00:08:36,929 --> 00:08:39,210
B_t is kind of like
square root of dt.
101
00:08:39,210 --> 00:08:40,860
It's not a good
notation, but if you
102
00:08:40,860 --> 00:08:45,870
do that-- these two terms are
significantly smaller than dt
103
00:08:45,870 --> 00:08:48,030
because you're
taking a power of it.
104
00:08:48,030 --> 00:08:51,905
dt square becomes a lot
smaller than dt, dt to the 3/2
105
00:08:51,905 --> 00:08:54,630
is a lot smaller than dt.
106
00:08:54,630 --> 00:08:59,601
But this one survives because
it's equal to dt here.
107
00:08:59,601 --> 00:09:01,225
That's just the
high-level description.
108
00:09:05,530 --> 00:09:08,320
That's a slightly more
sophisticated form
109
00:09:08,320 --> 00:09:09,470
of Ito's lemma.
110
00:09:09,470 --> 00:09:12,370
Let me write it down here.
111
00:09:12,370 --> 00:09:14,513
And let's just fix it now.
112
00:09:18,441 --> 00:09:48,880
If f of t of B_t-- that's d of
f is equal to-- Any questions?
113
00:09:58,610 --> 00:10:01,360
Just remember, from the
classical calculus term,
114
00:10:01,360 --> 00:10:05,385
we're only adding
this one term there.
115
00:10:05,385 --> 00:10:05,884
Yes?
116
00:10:05,884 --> 00:10:09,580
AUDIENCE: Why do
we have x there?
117
00:10:09,580 --> 00:10:15,400
PROFESSOR: Because the second
variable is supposed to be x.
118
00:10:15,400 --> 00:10:18,316
I don't want to write down
partial derivative with respect
119
00:10:18,316 --> 00:10:21,240
to a Brownian motion here
because it doesn't look good.
120
00:10:24,190 --> 00:10:26,490
It just means, take the
partial derivative with respect
121
00:10:26,490 --> 00:10:28,910
to the second term.
122
00:10:28,910 --> 00:10:33,340
So just view this as a
function f of t of x,
123
00:10:33,340 --> 00:10:42,080
evaluate it, and then plug
in x equal B_t in the end,
124
00:10:42,080 --> 00:10:44,250
because I don't want to
write down partial B_t here.
125
00:10:51,234 --> 00:10:51,900
Other questions?
126
00:11:11,810 --> 00:11:27,436
Consider a stochastic process
X of t such that d of X
127
00:11:27,436 --> 00:11:32,030
is equal to mu times d of t
plus sigma times d of B_t.
128
00:11:35,360 --> 00:11:38,160
This is almost like
a Brownian motion,
129
00:11:38,160 --> 00:11:39,720
but you have this
additional term.
130
00:11:39,720 --> 00:11:41,221
This is called a drift term.
131
00:11:46,130 --> 00:11:53,245
Basically, this happens if X_t
is equal to mu*t plus sigma
132
00:11:53,245 --> 00:11:55,710
of B_t.
133
00:11:55,710 --> 00:11:57,714
Mu and sigma are constants.
134
00:12:01,387 --> 00:12:02,970
From now on, what
we're going to study
135
00:12:02,970 --> 00:12:08,390
is stochastic process of
this type, whose difference
136
00:12:08,390 --> 00:12:12,608
can be written in terms of drift
term and the Brownian motion
137
00:12:12,608 --> 00:12:13,107
term.
138
00:12:16,100 --> 00:12:18,165
We want to do a slightly
more general form
139
00:12:18,165 --> 00:12:21,780
of Ito's lemma, where we
want f of t of X_t here.
140
00:12:25,502 --> 00:12:27,085
That will be the
main object of study.
141
00:12:31,593 --> 00:12:34,105
I'll finally state the
strongest Ito's lemma
142
00:12:34,105 --> 00:12:35,244
that we're going to use.
143
00:12:44,924 --> 00:12:54,370
f is some smooth function and
X_t is a stochastic process
144
00:12:54,370 --> 00:12:56,151
like that.
145
00:12:56,151 --> 00:12:57,095
X_t satisfies...
146
00:13:06,230 --> 00:13:08,770
where B_t is a Brownian motion.
147
00:13:08,770 --> 00:13:17,210
Then df of t, X_t
can be expressed
148
00:13:17,210 --> 00:13:38,510
as-- it's just getting
more and more complicated.
149
00:13:38,510 --> 00:13:41,580
But it's based on this one
simple principle, really.
150
00:13:41,580 --> 00:13:45,110
It all happened because
of quadratic variation.
151
00:13:45,110 --> 00:13:49,800
Now I'll show you why this form
deviates from this form when
152
00:13:49,800 --> 00:13:58,320
we replace B to an X.
153
00:13:58,320 --> 00:14:03,400
Remember here all other
terms didn't matter,
154
00:14:03,400 --> 00:14:08,595
that the only term that mattered
was partial square of f...
155
00:14:08,595 --> 00:14:11,888
of dx square.
156
00:14:17,490 --> 00:14:30,990
To prove this, note that df
is partial f over partial t
157
00:14:30,990 --> 00:14:37,066
dt plus partial f over
partial x d of X_t
158
00:14:37,066 --> 00:14:42,775
plus 1/2 of d of x squared.
159
00:14:45,490 --> 00:14:48,970
Just exactly the same, but I've
replaced the dB-- previously,
160
00:14:48,970 --> 00:14:52,885
what we had dB, I'm
replacing to dX.
161
00:14:52,885 --> 00:14:58,280
Now what changes is dX_t
can be written like that.
162
00:14:58,280 --> 00:15:03,580
If you just plug it
in, what you get here
163
00:15:03,580 --> 00:15:14,010
is partial f over partial
x mu dt plus sigma of dB_t.
164
00:15:14,010 --> 00:15:19,680
Then what you get here
is 1/2 of partials
165
00:15:19,680 --> 00:15:23,630
and then mu plus
sigma dB_t square.
166
00:15:26,620 --> 00:15:31,250
Out of those three terms here
we get mu square dt square
167
00:15:31,250 --> 00:15:37,590
plus 2 times mu sigma d mu dB
plus sigma square dB square.
168
00:15:37,590 --> 00:15:40,370
Only this was survives,
just as before.
169
00:15:40,370 --> 00:15:42,970
These ones disappear.
170
00:15:42,970 --> 00:15:45,180
And then you just
collect the terms.
171
00:15:45,180 --> 00:15:48,690
So dt-- there's one dt here.
172
00:15:48,690 --> 00:15:55,673
There's mu times that here,
and that one will become a dt.
173
00:15:55,673 --> 00:16:00,308
It's 1/2 of sigma
square partial square...
174
00:16:00,308 --> 00:16:01,300
of dt.
175
00:16:01,300 --> 00:16:04,770
And there's only
one dB_t term here.
176
00:16:04,770 --> 00:16:14,312
Sigma-- I made a mistake, sigma.
177
00:16:25,080 --> 00:16:27,480
This will be a form that
you'll use the most,
178
00:16:27,480 --> 00:16:32,710
because you want to evaluate
some stochastic process--
179
00:16:32,710 --> 00:16:36,150
some function that
depends on time
180
00:16:36,150 --> 00:16:37,431
and that stochastic process.
181
00:16:37,431 --> 00:16:39,180
You want to understand
the difference, df.
182
00:16:42,930 --> 00:16:44,660
The X would have
been written in terms
183
00:16:44,660 --> 00:16:47,000
of a Brownian motion
and a drift term,
184
00:16:47,000 --> 00:16:50,090
and then that's the
Ito lemma for you.
185
00:16:50,090 --> 00:16:51,680
But if you want to
just-- if you just
186
00:16:51,680 --> 00:16:56,420
see this for the first time,
it just looks too complicated.
187
00:16:56,420 --> 00:16:59,460
You don't understand where
all the terms are coming from.
188
00:16:59,460 --> 00:17:01,150
But in reality, what
it's really doing
189
00:17:01,150 --> 00:17:05,359
is just take this
Taylor expansion.
190
00:17:05,359 --> 00:17:08,170
Remember these two
classical terms,
191
00:17:08,170 --> 00:17:11,190
and remember that there's
one more term here.
192
00:17:11,190 --> 00:17:12,780
You can derive it
if you want to.
193
00:17:18,990 --> 00:17:21,030
Really try to know
where it all comes from.
194
00:17:21,030 --> 00:17:28,140
It all started from this one
fact, quadratic variation,
195
00:17:28,140 --> 00:17:32,940
because that made some of the
second derivative survive,
196
00:17:32,940 --> 00:17:34,850
and because of those,
you get these kind
197
00:17:34,850 --> 00:17:35,931
of complicated terms.
198
00:17:39,180 --> 00:17:39,680
Questions?
199
00:17:51,165 --> 00:17:53,444
Let's do some examples.
200
00:17:53,444 --> 00:17:54,110
That's too much.
201
00:18:02,390 --> 00:18:05,260
Sorry, I'm going to use it
a lot, so let me record it.
202
00:18:49,671 --> 00:18:54,590
Example number one.
203
00:18:54,590 --> 00:19:04,270
Let f of x be equal to
x square, and then you
204
00:19:04,270 --> 00:19:07,160
want to compute d of f at B_t.
205
00:19:13,590 --> 00:19:16,726
I'll give you three minutes
just to try a practice.
206
00:19:16,726 --> 00:19:18,030
Did you manage to do this?
207
00:19:25,610 --> 00:19:26,875
It's a very simple example.
208
00:19:32,400 --> 00:19:37,740
Assume it's just the
function of two variables,
209
00:19:37,740 --> 00:19:40,010
but it doesn't depend on t.
210
00:19:40,010 --> 00:19:44,030
You don't have to do that,
but let me just do that.
211
00:19:44,030 --> 00:19:45,755
Partial f over partial t is 0.
212
00:19:49,040 --> 00:19:52,970
Partial f over partial
x is equal to 2x,
213
00:19:52,970 --> 00:20:01,580
and the second derivative
equal to 2 at t, x.
214
00:20:01,580 --> 00:20:08,010
Now we just plug in
t comma B_t, and what
215
00:20:08,010 --> 00:20:11,345
you have is mu equals
0, sigma equals 1,
216
00:20:11,345 --> 00:20:13,011
if you want to write
it in this formula.
217
00:20:19,460 --> 00:20:25,940
What you're going to have
is 2 times B_t of dB_t
218
00:20:25,940 --> 00:20:27,490
plus 1 over 2 times 2dt.
219
00:20:30,322 --> 00:20:31,266
If you write it down.
220
00:20:34,570 --> 00:20:36,815
You can either use
these parameters
221
00:20:36,815 --> 00:20:41,200
and just plug in each of
them to figure it out.
222
00:20:41,200 --> 00:20:43,130
Or a different way
to do it is really
223
00:20:43,130 --> 00:20:45,340
write down, remember the proof.
224
00:20:45,340 --> 00:20:48,490
This is partial f
over partial t dt
225
00:20:48,490 --> 00:20:58,350
plus partial f over partial x
dx plus 1/2-- remember this one.
226
00:20:58,350 --> 00:21:00,045
And x is dB_t here.
227
00:21:04,190 --> 00:21:09,100
That one is 0, that one
was 2x, so 2B_t dB_t.
228
00:21:09,100 --> 00:21:11,872
Use it one more
time, so you get dt.
229
00:21:20,600 --> 00:21:21,280
Make sense?
230
00:21:24,160 --> 00:21:26,855
Let's do a few more examples.
231
00:22:03,150 --> 00:22:06,876
And you want to compute
d of f at t comma B of t.
232
00:22:11,280 --> 00:22:13,810
Let's do it this time.
233
00:22:13,810 --> 00:22:19,440
Again, partial f over
partial t dt plus partial f
234
00:22:19,440 --> 00:22:23,500
over partial x dB_t.
235
00:22:23,500 --> 00:22:24,820
That's the first-order terms.
236
00:22:24,820 --> 00:22:28,740
The second-order term
is 1/2 partial square f
237
00:22:28,740 --> 00:22:35,728
over partial x square of dB_t
square, which is equal to dt.
238
00:22:43,140 --> 00:22:43,640
Let's do it.
239
00:22:43,640 --> 00:22:48,720
Partial f over partial
t, you get mu times f.
240
00:22:48,720 --> 00:22:51,016
This one is just
equal to mu times f.
241
00:22:53,720 --> 00:22:55,220
Maybe I'm going too quick.
242
00:22:55,220 --> 00:23:02,510
Mu times e to the
mu t plus dx, dt.
243
00:23:02,510 --> 00:23:05,593
Partial f over partial
x is sigma times e
244
00:23:05,593 --> 00:23:12,120
to the mu t plus
dx, and then dB_t
245
00:23:12,120 --> 00:23:15,190
plus-- if you take
the second derivative,
246
00:23:15,190 --> 00:23:17,545
you do that again,
what you get is
247
00:23:17,545 --> 00:23:25,872
1/2, and then sigma square
times e to the mu t plus dx, dt.
248
00:23:25,872 --> 00:23:26,372
Yes?
249
00:23:26,372 --> 00:23:28,012
AUDIENCE: In the original
equation that you just wrote,
250
00:23:28,012 --> 00:23:29,816
isn't it 1/2 times
sigma squared,
251
00:23:29,816 --> 00:23:31,784
and then the second derivative?
252
00:23:31,784 --> 00:23:33,854
Up there.
253
00:23:33,854 --> 00:23:34,520
PROFESSOR: Here?
254
00:23:34,520 --> 00:23:35,470
AUDIENCE: Yes.
255
00:23:35,470 --> 00:23:36,094
PROFESSOR: 1/2?
256
00:23:36,094 --> 00:23:37,940
AUDIENCE: Times sigma squared.
257
00:23:37,940 --> 00:23:40,510
PROFESSOR: Oh, sigma-- OK,
that's a good question.
258
00:23:40,510 --> 00:23:44,070
But that sigma is different.
259
00:23:44,070 --> 00:23:46,250
That's if you plug in X_t here.
260
00:23:46,250 --> 00:23:50,790
If you plug in X_t
where X_t is equal to mu
261
00:23:50,790 --> 00:23:57,170
prime dt plus sigma
prime d of B_t,
262
00:23:57,170 --> 00:23:59,500
then that sigma prime will
become a sigma prime square
263
00:23:59,500 --> 00:24:02,020
here.
264
00:24:02,020 --> 00:24:04,124
But here the function
is mu and sigma,
265
00:24:04,124 --> 00:24:05,540
so maybe it's not
a good notation.
266
00:24:05,540 --> 00:24:07,383
Let me use a and b here instead.
267
00:24:11,170 --> 00:24:13,890
The sigma here is
different from here.
268
00:24:13,890 --> 00:24:15,950
AUDIENCE: Yeah, that
makes a lot more sense.
269
00:24:15,950 --> 00:24:18,826
PROFESSOR: If you
replace a and b,
270
00:24:18,826 --> 00:24:23,310
but I already wrote down
all mu's and sigma's.
271
00:24:23,310 --> 00:24:25,495
That's a good point, actually.
272
00:24:25,495 --> 00:24:26,995
But that's when you
want to consider
273
00:24:26,995 --> 00:24:29,250
a general stochastic
process here
274
00:24:29,250 --> 00:24:31,276
other than Brownian motion.
275
00:24:31,276 --> 00:24:33,030
But here it's just
a Brownian motion,
276
00:24:33,030 --> 00:24:35,376
so it's the most simple form.
277
00:24:35,376 --> 00:24:36,375
And that's what you get.
278
00:24:39,300 --> 00:24:45,940
Mu plus 1/2 sigma square-- and
these are just all f itself.
279
00:24:45,940 --> 00:24:48,130
That's the good thing
about exponential.
280
00:24:48,130 --> 00:24:51,939
f times dt plus
sigma times d of B_t.
281
00:25:03,930 --> 00:25:04,570
Make sense?
282
00:25:14,180 --> 00:25:17,896
And there's a reason I
was covering this example.
283
00:25:17,896 --> 00:25:22,140
It's because-- let's come
back to this question.
284
00:25:22,140 --> 00:25:35,110
You want to model stock price
using Brownian motion, Brownian
285
00:25:35,110 --> 00:25:40,760
process, S of t.
286
00:25:40,760 --> 00:25:43,040
But you don't want S_t
to be a Brownian motion.
287
00:25:43,040 --> 00:25:46,760
What you want is a
percentile difference
288
00:25:46,760 --> 00:25:51,400
to be a Brownian motion, so you
want this percentile difference
289
00:25:51,400 --> 00:25:58,505
to behave like a Brownian
motion with some variance.
290
00:26:03,350 --> 00:26:11,500
The question was, is S_t equal
to e to the sigma times B of t
291
00:26:11,500 --> 00:26:12,745
in this case?
292
00:26:12,745 --> 00:26:16,260
And I already told you last
time that no, it's not true.
293
00:26:16,260 --> 00:26:18,360
We can now see
why it's not true.
294
00:26:18,360 --> 00:26:20,880
Take this function, S_t
equals e to the sigma
295
00:26:20,880 --> 00:26:24,990
B_t, that's exactly where
mu is equal to 0 here.
296
00:26:24,990 --> 00:26:30,590
What we got here was d of S_t,
in this case, is equal to-- mu
297
00:26:30,590 --> 00:26:36,430
is 0, so we get 1/2 of sigma
square times dt plus sigma
298
00:26:36,430 --> 00:26:37,274
times d of B_t.
299
00:26:40,180 --> 00:26:44,670
We originally were
targeting sigma times dB_t,
300
00:26:44,670 --> 00:26:47,890
but we got this
additional term which we
301
00:26:47,890 --> 00:26:51,112
didn't want in the first hand.
302
00:26:51,112 --> 00:26:52,570
In other words, we
have this drift.
303
00:27:01,455 --> 00:27:03,080
I wasn't really clear
in the beginning,
304
00:27:03,080 --> 00:27:05,850
but our goal was to
model stock price
305
00:27:05,850 --> 00:27:10,920
where the expected
value is 0 at all times.
306
00:27:10,920 --> 00:27:12,970
Our guess was to take
e to the sigma B_t,
307
00:27:12,970 --> 00:27:14,890
but it turns out
that in this case
308
00:27:14,890 --> 00:27:16,870
we have a drift,
if you just take
309
00:27:16,870 --> 00:27:19,570
natural e to the sigma of B_t.
310
00:27:19,570 --> 00:27:21,230
To remove that drift,
what you can do
311
00:27:21,230 --> 00:27:23,350
is subtract that term somehow.
312
00:27:23,350 --> 00:27:26,570
If you can get rid of that
term then you can see,
313
00:27:26,570 --> 00:27:30,120
if you add this mu to be
minus 1 over 2 sigma square,
314
00:27:30,120 --> 00:27:31,340
you can remove that term.
315
00:27:34,460 --> 00:27:35,650
That's why it doesn't work.
316
00:27:35,650 --> 00:27:47,570
So instead use S of t equals
e to the minus 1 over 2 sigma
317
00:27:47,570 --> 00:27:52,532
square t plus sigma of B_t.
318
00:27:58,330 --> 00:28:02,850
That's the geometric Brownian
motion without drift.
319
00:28:02,850 --> 00:28:05,820
And the reason it has no
drift is because of that.
320
00:28:05,820 --> 00:28:07,300
If you actually do
the computation,
321
00:28:07,300 --> 00:28:08,390
the dt term disappears.
322
00:28:28,911 --> 00:28:29,410
Question?
323
00:28:35,580 --> 00:28:39,000
So far we have been
discussing differentiation.
324
00:28:39,000 --> 00:28:40,706
Now let's talk
about integration.
325
00:28:40,706 --> 00:28:41,206
Yes?
326
00:28:41,206 --> 00:28:47,122
AUDIENCE: Could you we do get
this solution as [INAUDIBLE].
327
00:28:47,122 --> 00:28:50,573
Could you also
describe what it means?
328
00:28:50,573 --> 00:28:55,996
What does it mean,
this solution of B_t?
329
00:28:55,996 --> 00:28:58,461
Does that mean if we
have a sample path B_t,
330
00:28:58,461 --> 00:29:01,440
then we could get a
sample path for S_t?
331
00:29:01,440 --> 00:29:03,670
PROFESSOR: Oh, what
this means, yes.
332
00:29:03,670 --> 00:29:07,360
Whenever you have the B_t
value, just at each time
333
00:29:07,360 --> 00:29:10,050
take the exponential value.
334
00:29:10,050 --> 00:29:13,460
Because-- why we want to express
this in terms of a Brownian
335
00:29:13,460 --> 00:29:14,940
motion is, for
Brownian motion we
336
00:29:14,940 --> 00:29:17,030
have a pretty good
understanding.
337
00:29:17,030 --> 00:29:21,280
It's a really good process
you understand fairly well,
338
00:29:21,280 --> 00:29:25,200
and you have good control on it.
339
00:29:25,200 --> 00:29:28,840
But the problem is you want to
have a process whose percentile
340
00:29:28,840 --> 00:29:31,990
difference behaves
like a Brownian motion.
341
00:29:31,990 --> 00:29:34,340
And this gives you a
way of describing it
342
00:29:34,340 --> 00:29:37,290
in terms of Brownian motion, as
an exponential function of it.
343
00:29:43,390 --> 00:29:46,555
Does that answer your question?
344
00:29:46,555 --> 00:29:48,330
AUDIENCE: Right,
distribution means
345
00:29:48,330 --> 00:29:50,720
that if we have a
sample path B_t,
346
00:29:50,720 --> 00:29:54,544
that would be the corresponding
sample path for S of t?
347
00:29:54,544 --> 00:29:58,260
Is it a pointwise evaluation?
348
00:29:58,260 --> 00:30:00,270
PROFESSOR: That's a
good question, actually.
349
00:30:00,270 --> 00:30:02,950
Think of it as a
pointwise evaluation.
350
00:30:02,950 --> 00:30:07,400
That is not always
correct, but for most
351
00:30:07,400 --> 00:30:09,150
of the things that
we will cover,
352
00:30:09,150 --> 00:30:13,120
it's safe to think
about it that way.
353
00:30:13,120 --> 00:30:16,690
But if you think about it
path-wise all the time,
354
00:30:16,690 --> 00:30:18,680
eventually it fails.
355
00:30:18,680 --> 00:30:20,187
But that's a very
advanced topic.
356
00:30:32,130 --> 00:30:33,740
So what this question
is, basically
357
00:30:33,740 --> 00:30:37,350
B_t is a probability space,
it's a probability distribution
358
00:30:37,350 --> 00:30:39,390
over paths.
359
00:30:39,390 --> 00:30:43,060
For this equation, if you just
look at it, it looks right,
360
00:30:43,060 --> 00:30:44,650
but it doesn't
really make sense,
361
00:30:44,650 --> 00:30:46,964
because B_t-- if it's a
probability distribution, what
362
00:30:46,964 --> 00:30:47,630
is e to the B_t?
363
00:30:50,450 --> 00:30:52,530
Basically, what
it's saying is B_t
364
00:30:52,530 --> 00:30:55,210
is a probability
distribution over paths.
365
00:30:55,210 --> 00:30:58,700
If you take omega according
to-- a path according
366
00:30:58,700 --> 00:31:02,890
to the Brownian motion sample
probability distribution,
367
00:31:02,890 --> 00:31:08,230
and for this path it's well
defined, this function.
368
00:31:08,230 --> 00:31:13,700
So the probability density
function of this path
369
00:31:13,700 --> 00:31:16,910
is equal to the probability
density function of e
370
00:31:16,910 --> 00:31:19,435
to the whatever that is
in this distribution.
371
00:31:24,410 --> 00:31:26,300
Maybe it confused you more.
372
00:31:26,300 --> 00:31:30,009
Just consider this as some path,
some well-defined function,
373
00:31:30,009 --> 00:31:31,550
and you have a
well-defined function.
374
00:31:39,490 --> 00:31:40,853
Integral definition.
375
00:31:44,000 --> 00:31:46,510
I will first give you a
very, very stupid definition
376
00:31:46,510 --> 00:31:49,340
of integration.
377
00:31:49,340 --> 00:32:00,293
We say that we define
F as the integration...
378
00:32:12,270 --> 00:32:25,574
if d of F is equal
to f dB_t plus-- we
379
00:32:25,574 --> 00:32:27,365
define it as an inverse
of differentiation.
380
00:32:30,200 --> 00:32:34,860
Because differentiation
is now well-defined--
381
00:32:34,860 --> 00:32:39,690
we just defined integration
as the inverse of it,
382
00:32:39,690 --> 00:32:42,170
just as in classical calculus.
383
00:32:46,030 --> 00:32:47,810
So far, it doesn't
have that good meaning,
384
00:32:47,810 --> 00:32:51,160
other than being
an inverse of it,
385
00:32:51,160 --> 00:32:52,710
but at least it's well-defined.
386
00:32:52,710 --> 00:32:54,780
The question is, does it exist?
387
00:32:54,780 --> 00:32:57,445
Given f and g, does it exist,
does integration always exist,
388
00:32:57,445 --> 00:32:58,060
and so on.
389
00:32:58,060 --> 00:32:59,690
There's lots of
questions to ask,
390
00:32:59,690 --> 00:33:02,570
but at least this
is some definition.
391
00:33:02,570 --> 00:33:11,760
And the natural question is,
does there exist a Riemannian
392
00:33:11,760 --> 00:33:12,740
sum type description?
393
00:33:25,430 --> 00:33:28,580
That means-- if you remember
how we defined integral
394
00:33:28,580 --> 00:33:47,740
in calculus, you have a
function f, integration
395
00:33:47,740 --> 00:33:58,570
of f from a to b according to
the Riemannian sum description
396
00:33:58,570 --> 00:34:04,660
was, you just chop the
interval into very fine pieces,
397
00:34:04,660 --> 00:34:07,730
a_0, a_1, a_2, a_3,
dot, dot, dot--
398
00:34:07,730 --> 00:34:16,750
and then sum the area of these
boxes, and take the limit.
399
00:34:16,750 --> 00:34:20,570
And this is the limit
of Riemannian sums.
400
00:34:26,420 --> 00:34:33,920
Slightly more, if you want, it's
the limit as n goes to infinity
401
00:34:33,920 --> 00:34:40,790
of the function 1 over n times
the sum of i equal zero to t--
402
00:34:40,790 --> 00:34:48,587
I'll just call it 0 to b-- f of
t*b over n minus f of t minus 1
403
00:34:48,587 --> 00:34:51,461
over n.
404
00:34:51,461 --> 00:34:52,734
Does this ring a bell?
405
00:35:03,787 --> 00:35:04,287
Question?
406
00:35:04,287 --> 00:35:05,162
AUDIENCE: [INAUDIBLE]
407
00:35:10,780 --> 00:35:13,832
PROFESSOR: No, you're right.
408
00:35:13,832 --> 00:35:17,115
Good point, no we don't.
409
00:35:17,115 --> 00:35:17,615
Thanks.
410
00:35:22,570 --> 00:35:26,010
Does integral
defined in this way
411
00:35:26,010 --> 00:35:31,390
have this Riemannian sum type
description, is the question.
412
00:35:31,390 --> 00:35:33,110
So keep that in mind.
413
00:35:33,110 --> 00:35:36,660
I will come back to
this point later.
414
00:35:36,660 --> 00:35:41,839
In fact, it turns out to
be a very deep question
415
00:35:41,839 --> 00:35:43,630
and very important
question, this question,
416
00:35:43,630 --> 00:35:48,850
because if you remember
like I hope you remember,
417
00:35:48,850 --> 00:35:51,860
in the Riemannian sum, it
didn't matter which point you
418
00:35:51,860 --> 00:35:54,610
took in this interval.
419
00:35:54,610 --> 00:35:56,910
That was the whole point.
420
00:35:56,910 --> 00:35:58,780
You have the function.
421
00:35:58,780 --> 00:36:02,430
In the interval a_i to
a_(i+1), you take any point
422
00:36:02,430 --> 00:36:07,610
in the middle and make a
rectangle according to that
423
00:36:07,610 --> 00:36:08,642
point.
424
00:36:08,642 --> 00:36:10,350
And then, no matter
which point you take,
425
00:36:10,350 --> 00:36:12,740
when you go to the limit,
you had exactly the same sum
426
00:36:12,740 --> 00:36:14,190
all the time.
427
00:36:14,190 --> 00:36:16,560
That's how you define the limit.
428
00:36:16,560 --> 00:36:20,960
But what's really
interesting here
429
00:36:20,960 --> 00:36:24,090
is that it's no longer true.
430
00:36:24,090 --> 00:36:26,000
If you take the left
point all the time,
431
00:36:26,000 --> 00:36:28,110
and you take the right
point all the time,
432
00:36:28,110 --> 00:36:30,570
the two limits are different.
433
00:36:30,570 --> 00:36:33,020
And again, that's due to
the quadratic variation,
434
00:36:33,020 --> 00:36:38,980
because that much of variance
can accumulate over time.
435
00:36:38,980 --> 00:36:42,560
That's the reason we didn't
start with Riemannian sum type
436
00:36:42,560 --> 00:36:44,450
definition of integral.
437
00:36:44,450 --> 00:36:47,490
But I'll just make one remark.
438
00:36:47,490 --> 00:37:01,220
Ito integral is the
limit of Riemannian sums
439
00:37:01,220 --> 00:37:08,240
when always take the leftmost
point of each interval.
440
00:37:16,700 --> 00:37:20,670
So you chop down this curve
at-- the time interval
441
00:37:20,670 --> 00:37:23,106
into pieces, and
for each rectangle,
442
00:37:23,106 --> 00:37:25,230
pick the leftmost point,
and use it as a rectangle.
443
00:37:30,722 --> 00:37:31,680
And you take the limit.
444
00:37:31,680 --> 00:37:33,570
That will be your
Ito integral defined.
445
00:37:33,570 --> 00:37:37,070
It will be exactly equal to this
thing, the inverse of our Ito
446
00:37:37,070 --> 00:37:38,754
differentiation.
447
00:37:38,754 --> 00:37:40,170
I won't be able
to go into detail.
448
00:37:40,170 --> 00:37:44,080
What's more
interesting is instead,
449
00:37:44,080 --> 00:37:47,170
what happens if you take the
rightmost point all the time,
450
00:37:47,170 --> 00:37:51,680
you get an equivalent
theory of calculus.
451
00:37:51,680 --> 00:37:53,380
It's just like Ito's calculus.
452
00:37:53,380 --> 00:37:57,160
It looks really, really similar
and it's coherent itself,
453
00:37:57,160 --> 00:37:59,170
so there is no
logical flaw in it.
454
00:37:59,170 --> 00:38:01,260
It all makes sense,
but the only difference
455
00:38:01,260 --> 00:38:03,984
is instead of a plus in
the second-order term,
456
00:38:03,984 --> 00:38:04,650
you get minuses.
457
00:38:07,500 --> 00:38:09,940
Let me just make this
remark, because it's just
458
00:38:09,940 --> 00:38:15,350
a theoretical part, this thing,
but I think it's really cool.
459
00:38:15,350 --> 00:38:22,150
Remark-- there's this
and equivalent version.
460
00:38:22,150 --> 00:38:24,430
Maybe equivalent is
not the right word,
461
00:38:24,430 --> 00:38:26,820
but a very similar
version of Ito
462
00:38:26,820 --> 00:38:33,000
calculus such that
basically, what
463
00:38:33,000 --> 00:38:38,510
it says is d B_t square
is equal to minus dt.
464
00:38:38,510 --> 00:38:40,320
Then that changed
a lot of things.
465
00:38:40,320 --> 00:38:44,510
But this part, it's
not that important.
466
00:38:44,510 --> 00:38:48,740
Just cool stuff.
467
00:38:48,740 --> 00:38:53,410
Let's think about this a
little bit more, this fact.
468
00:38:53,410 --> 00:38:55,970
Taking the leftmost
point all the time
469
00:38:55,970 --> 00:38:59,820
means if you want to make
a decision for your time
470
00:38:59,820 --> 00:39:05,630
interval-- so at time t of
i and time t of i plus 1,
471
00:39:05,630 --> 00:39:08,760
let's say it's the stock price.
472
00:39:08,760 --> 00:39:14,590
You want to say that you had
so many stocks in this time
473
00:39:14,590 --> 00:39:16,370
interval.
474
00:39:16,370 --> 00:39:20,400
Let's say you had so many
stocks in this time interval
475
00:39:20,400 --> 00:39:23,430
according to the values
between this and this.
476
00:39:23,430 --> 00:39:26,660
In real world, your only
choice you have is you
477
00:39:26,660 --> 00:39:30,700
have to make the
decision at time t of i.
478
00:39:30,700 --> 00:39:33,580
Your choice cannot depend
on the future time.
479
00:39:33,580 --> 00:39:36,650
You can't suddenly say, OK,
in this interval the stock
480
00:39:36,650 --> 00:39:38,420
price increased a
lot, so I'll assume
481
00:39:38,420 --> 00:39:42,930
that I had a lot of
stocks in this interval.
482
00:39:42,930 --> 00:39:46,190
In this interval, I knew
it was going to drop,
483
00:39:46,190 --> 00:39:50,060
so I'll just take the
rightmost interval.
484
00:39:50,060 --> 00:39:52,410
I'll assume that I only
had this many stock.
485
00:39:52,410 --> 00:39:53,460
You can't do that.
486
00:39:53,460 --> 00:39:56,850
Your decision has to be
based on the leftmost point,
487
00:39:56,850 --> 00:39:58,630
because the time.
488
00:39:58,630 --> 00:40:00,880
You can't see the future.
489
00:40:00,880 --> 00:40:04,690
And the reason Ito's calculus
works well in our setting is
490
00:40:04,690 --> 00:40:09,380
because of this fact, because it
has inside it the fact that you
491
00:40:09,380 --> 00:40:10,820
cannot see the future.
492
00:40:10,820 --> 00:40:16,310
Every decision is made
based on the leftmost time.
493
00:40:16,310 --> 00:40:18,845
If you want to make a decision
for your time interval,
494
00:40:18,845 --> 00:40:21,700
you have to do it
in the beginning.
495
00:40:21,700 --> 00:40:27,240
That intuition is hidden
inside of the theory,
496
00:40:27,240 --> 00:40:29,390
and that's why it works so well.
497
00:40:29,390 --> 00:40:33,450
Let me reiterate this
part a little bit more.
498
00:40:33,450 --> 00:40:36,500
It's the definition
of these things
499
00:40:36,500 --> 00:40:39,540
where you're only
allowed to-- at time t,
500
00:40:39,540 --> 00:40:42,230
you're only allowed to use
the information up to time t.
501
00:40:54,390 --> 00:41:26,010
Definition: delta t is an
adapted process-- sorry,
502
00:41:26,010 --> 00:41:29,730
adapted to another
stochastic process X_t--
503
00:41:29,730 --> 00:41:38,920
if for all values
of time variables
504
00:41:38,920 --> 00:41:48,110
delta t depends only
on X_0 up to X_t.
505
00:41:50,930 --> 00:41:53,360
There's a lot of vague
statements inside here,
506
00:41:53,360 --> 00:41:55,120
but what I'm trying
to say is just
507
00:41:55,120 --> 00:41:59,947
assume X is the Brownian
motion underlying stock price.
508
00:41:59,947 --> 00:42:00,905
Your stock is changing.
509
00:42:04,500 --> 00:42:06,305
You want to come
up with a strategy,
510
00:42:06,305 --> 00:42:08,270
and you want to say
that mathematically
511
00:42:08,270 --> 00:42:11,280
this strategy makes sense.
512
00:42:11,280 --> 00:42:13,220
And what it's saying
is if your strategy
513
00:42:13,220 --> 00:42:17,050
makes your decision
at time t is only
514
00:42:17,050 --> 00:42:19,840
based on the past values
of your stock price,
515
00:42:19,840 --> 00:42:23,750
then that's an adapted process.
516
00:42:23,750 --> 00:42:26,590
This defines the processes
that are reasonable,
517
00:42:26,590 --> 00:42:28,540
that cannot see future.
518
00:42:28,540 --> 00:42:31,030
And these are all--
in terms of strategy,
519
00:42:31,030 --> 00:42:34,400
if delta_t is a
portfolio strategy,
520
00:42:34,400 --> 00:42:37,262
these are the only meaningful
strategies that you can use.
521
00:42:40,240 --> 00:42:42,740
And because of what I said
before, because we're always
522
00:42:42,740 --> 00:42:45,580
taking the leftmost
point, adaptive
523
00:42:45,580 --> 00:42:50,440
processes just also fit very
well with Ito's calculus.
524
00:42:50,440 --> 00:42:53,050
They'll come into
play altogether.
525
00:42:55,670 --> 00:42:56,655
Just a few examples.
526
00:43:10,445 --> 00:43:13,740
First, a very stupid example.
527
00:43:13,740 --> 00:43:15,290
X_t is adapted to X_t.
528
00:43:20,170 --> 00:43:23,230
Of course, because
at time, X_t really
529
00:43:23,230 --> 00:43:26,710
depends on only
X_t, nothing else.
530
00:43:26,710 --> 00:43:36,060
Two, X_(t+1) is
not adapted to X_t.
531
00:43:36,060 --> 00:43:37,580
This is maybe a
little bit vague,
532
00:43:37,580 --> 00:43:41,412
so we'll call it
Y_t equals X_(t+1).
533
00:43:44,090 --> 00:43:49,240
Y_t is the value at t
plus 1, and it's not based
534
00:43:49,240 --> 00:43:50,850
on the values up to time t.
535
00:43:50,850 --> 00:43:52,600
Just a very artificial example.
536
00:43:56,405 --> 00:44:03,178
Another example, delta
t equals minimum...
537
00:44:06,136 --> 00:44:07,122
is adapted.
538
00:44:21,419 --> 00:44:23,180
And I'll let you think about it.
539
00:44:23,180 --> 00:44:24,850
The fourth is quite interesting.
540
00:44:24,850 --> 00:44:27,540
Suppose T is fixed,
some large integer,
541
00:44:27,540 --> 00:44:30,140
or some large real number.
542
00:44:30,140 --> 00:44:44,170
Then you let delta t to be the
maximum where X of s, where...
543
00:44:50,600 --> 00:44:51,370
It's not adapted.
544
00:44:58,790 --> 00:44:59,455
What is this?
545
00:44:59,455 --> 00:45:02,020
This means at time T,
I'm going to take at it
546
00:45:02,020 --> 00:45:08,850
this value, the
maximum of all value
547
00:45:08,850 --> 00:45:11,340
inside this part, the future.
548
00:45:11,340 --> 00:45:13,469
This refers to the future.
549
00:45:13,469 --> 00:45:14,635
It's not an adapted process.
550
00:45:21,637 --> 00:45:22,220
Any questions?
551
00:45:25,290 --> 00:45:28,340
Now we're ready to talk
about the properties
552
00:45:28,340 --> 00:45:31,190
of Ito's integral.
553
00:45:31,190 --> 00:45:34,340
Let's quickly
review what we have.
554
00:45:34,340 --> 00:45:38,380
First, I defined Ito's lemma--
that means differentiation
555
00:45:38,380 --> 00:45:41,250
in Ito calculus.
556
00:45:41,250 --> 00:45:45,080
Then I defined integration using
differentiation-- integration
557
00:45:45,080 --> 00:45:48,020
was an inverse operation
of the differentiation.
558
00:45:48,020 --> 00:45:50,500
But this integration also had
an alternative description
559
00:45:50,500 --> 00:45:53,260
in terms of
Riemannian sums, where
560
00:45:53,260 --> 00:45:58,650
you're taking just the
leftmost point as the reference
561
00:45:58,650 --> 00:46:01,700
point for each interval.
562
00:46:01,700 --> 00:46:04,370
And then, as you
see, this naturally
563
00:46:04,370 --> 00:46:08,090
had this concept of
using the leftmost point.
564
00:46:08,090 --> 00:46:12,180
And to abstract
that concept, we've
565
00:46:12,180 --> 00:46:15,660
come up with this adapted
process, very natural process,
566
00:46:15,660 --> 00:46:17,710
which is like the
real-life procedures,
567
00:46:17,710 --> 00:46:20,900
real-life strategies
we can think of.
568
00:46:20,900 --> 00:46:22,700
Now let's see what
happens when you
569
00:46:22,700 --> 00:46:25,442
take the integral of
adapted processes.
570
00:46:25,442 --> 00:46:27,870
Ito integral has
really cool properties.
571
00:46:59,540 --> 00:47:03,000
The first thing is about
normal distribution.
572
00:47:03,000 --> 00:47:08,840
B_t has normal
distribution of 0 up to t.
573
00:47:08,840 --> 00:47:11,170
So your Brownian
motion at time t
574
00:47:11,170 --> 00:47:13,780
has normal
distribution with 0, t.
575
00:47:13,780 --> 00:47:17,090
That means if your stochastic
process is some constant times
576
00:47:17,090 --> 00:47:23,540
B of t, of course, then
you have 0 and c square t.
577
00:47:23,540 --> 00:47:26,780
It's still a normal variable.
578
00:47:26,780 --> 00:47:28,770
That means if you
integrate, that's
579
00:47:28,770 --> 00:47:31,130
the integration of some sigma.
580
00:47:39,878 --> 00:47:42,058
That's the integration
of sigma of dB_t.
581
00:47:46,680 --> 00:47:49,280
If sigma is a fixed
constant, when
582
00:47:49,280 --> 00:47:52,830
you take the Ito
integral of sigma times
583
00:47:52,830 --> 00:47:55,200
dB_t, this constant,
at each time
584
00:47:55,200 --> 00:47:58,210
you get a normal distribution.
585
00:47:58,210 --> 00:48:00,550
And this is like saying the
sum of normal distribution
586
00:48:00,550 --> 00:48:02,330
is also normal distribution.
587
00:48:02,330 --> 00:48:04,090
It has this hidden
fact, because integral
588
00:48:04,090 --> 00:48:06,980
is like sum in the limit.
589
00:48:06,980 --> 00:48:10,456
And this can be generalized.
590
00:48:10,456 --> 00:48:18,560
If delta t is a process
depending only on the time
591
00:48:18,560 --> 00:48:27,660
variable-- so it does not depend
on the Brownian motion-- then
592
00:48:27,660 --> 00:48:35,810
the process X of t equals the
integration of delta t dB_t
593
00:48:35,810 --> 00:48:50,420
has normal distribution at
all time, just like this.
594
00:48:50,420 --> 00:48:52,580
We don't know the
exact variance yet;
595
00:48:52,580 --> 00:48:55,280
the variance will
depend on the sigmas.
596
00:48:55,280 --> 00:48:57,334
But still, it's like a
sum of normal variables,
597
00:48:57,334 --> 00:48:58,750
so we'll have
normal distribution.
598
00:49:03,490 --> 00:49:05,360
In fact, it just gets
better and better.
599
00:49:10,140 --> 00:49:14,950
The second fact is
called Ito isometry.
600
00:49:14,950 --> 00:49:15,930
That was cool.
601
00:49:15,930 --> 00:49:17,136
Can we compute the variance?
602
00:49:29,611 --> 00:49:30,110
Yes?
603
00:49:30,110 --> 00:49:31,466
AUDIENCE: Can you
put that board up?
604
00:49:31,466 --> 00:49:32,132
PROFESSOR: Sure.
605
00:49:34,630 --> 00:49:35,970
AUDIENCE: Does it go up?
606
00:49:35,970 --> 00:49:37,900
PROFESSOR: This
one doesn't go up.
607
00:49:37,900 --> 00:49:40,070
That's bad.
608
00:49:40,070 --> 00:49:41,240
I wish it did go up.
609
00:49:49,020 --> 00:49:52,060
This has a name
called Ito isometry.
610
00:49:56,740 --> 00:49:58,890
Can be used to
compute the variance.
611
00:49:58,890 --> 00:50:01,640
B_t is a Brownian
motion, delta t
612
00:50:01,640 --> 00:50:03,480
is adapted to a Brownian motion.
613
00:50:09,050 --> 00:50:17,610
Then the expectation
of your Ito integral--
614
00:50:17,610 --> 00:50:21,600
that's the Ito integral
of your adapted process.
615
00:50:21,600 --> 00:50:25,440
That's the variance-- we
take the square of it--
616
00:50:25,440 --> 00:50:29,847
is equal to something cool.
617
00:50:36,180 --> 00:50:38,456
The square just comes in.
618
00:50:38,456 --> 00:50:39,500
Quite nice, isn't it?
619
00:50:44,520 --> 00:50:48,400
I won't prove it, but
let me tell you why.
620
00:50:48,400 --> 00:50:50,300
We already saw this
phenomenon before.
621
00:50:50,300 --> 00:50:51,960
This is basically
quadratic variation.
622
00:50:58,000 --> 00:50:59,560
And the proof also uses it.
623
00:50:59,560 --> 00:51:03,160
If you take delta s
equals to 1-- sorry,
624
00:51:03,160 --> 00:51:09,680
I was using Korean-- 1 at all
time, then what we have is
625
00:51:09,680 --> 00:51:13,490
here you get a
Brownian motion, B_t.
626
00:51:13,490 --> 00:51:19,530
So on the left you get like
expectation of B_t square,
627
00:51:19,530 --> 00:51:21,525
and on the right,
what you get is t.
628
00:51:24,440 --> 00:51:27,445
Because when delta
s is equal to 1
629
00:51:27,445 --> 00:51:30,260
at all time, when you have
to get from 0 to t you get t,
630
00:51:30,260 --> 00:51:32,730
and you have t on
the right hand side.
631
00:51:32,730 --> 00:51:35,180
That's what it's saying.
632
00:51:35,180 --> 00:51:37,455
And that was the content
of quadratic variation,
633
00:51:37,455 --> 00:51:38,840
if you remember.
634
00:51:38,840 --> 00:51:42,484
We're summing the squares--
maybe not exactly this,
635
00:51:42,484 --> 00:51:44,650
but you're summing the
squares over small intervals.
636
00:52:00,530 --> 00:52:02,510
So that's a really
good fact that you can
637
00:52:02,510 --> 00:52:05,900
use to compute the variance.
638
00:52:05,900 --> 00:52:08,220
You have an Ito integral,
you know the square,
639
00:52:08,220 --> 00:52:10,190
can be computed this simple way.
640
00:52:14,110 --> 00:52:15,090
That's really cool.
641
00:52:17,850 --> 00:52:19,080
And one more property.
642
00:52:19,080 --> 00:52:22,560
This one will be
really important.
643
00:52:22,560 --> 00:52:24,295
You'll see it a lot
in future lectures.
644
00:52:28,630 --> 00:52:31,190
It's that when is Ito
integral a martingale?
645
00:52:46,630 --> 00:52:48,030
What's a martingale?
646
00:52:48,030 --> 00:52:52,310
Martingale meant if you
have a stochastic process,
647
00:52:52,310 --> 00:53:01,080
at any time t, whatever happens
after that, the expected value
648
00:53:01,080 --> 00:53:03,890
at time t is equal to 0.
649
00:53:03,890 --> 00:53:07,630
It doesn't have any natural
tendency to go up or go down.
650
00:53:07,630 --> 00:53:10,357
No matter which point
you stop your process
651
00:53:10,357 --> 00:53:12,815
and you see your future, it
doesn't have a natural tendency
652
00:53:12,815 --> 00:53:15,470
to go up or go down.
653
00:53:15,470 --> 00:53:29,190
In formal language, it can
be defined as where F_t
654
00:53:29,190 --> 00:53:32,670
is the events X_0 up to X_t.
655
00:53:35,890 --> 00:53:39,610
So if you take the
conditional expectation
656
00:53:39,610 --> 00:53:42,300
based on whatever
happened up to time t,
657
00:53:42,300 --> 00:53:44,235
that expectation will
just be whatever value
658
00:53:44,235 --> 00:53:45,384
you have at that time.
659
00:53:48,524 --> 00:53:51,190
Intuitively, that just means you
don't have any natural tendency
660
00:53:51,190 --> 00:53:53,900
to go up or go down.
661
00:53:53,900 --> 00:53:59,470
Question is, when is an
Ito integral a martingale?
662
00:54:28,985 --> 00:54:35,710
Adapted to B of t, then...
663
00:54:45,344 --> 00:54:46,010
is a martingale.
664
00:54:51,030 --> 00:54:54,090
As long as g is not
some crazy function,
665
00:54:54,090 --> 00:55:05,392
as long as g is reasonable--
one way can be reasonable if its
666
00:55:05,392 --> 00:55:07,900
L^2-norm is bounded.
667
00:55:07,900 --> 00:55:11,540
If you don't know what it
means, you can safely ignore it.
668
00:55:19,030 --> 00:55:23,490
Basically, if g doesn't-- it's
not a crazy function if it
669
00:55:23,490 --> 00:55:27,800
doesn't grow too fast, then
in most cases this integral is
670
00:55:27,800 --> 00:55:28,960
always a martingale.
671
00:55:31,590 --> 00:55:34,800
If you flip it--
remember, integral
672
00:55:34,800 --> 00:55:38,880
was defined as the inverse
of differentiation.
673
00:55:38,880 --> 00:55:42,700
So if dX_t is equal to
some function mu, that
674
00:55:42,700 --> 00:55:48,631
depends on both t and
B_t, times dt, plus sigma
675
00:55:48,631 --> 00:55:56,940
of dB_t, what this means
is X_t is a martingale
676
00:55:56,940 --> 00:56:02,690
if that is 0 at
all time, always.
677
00:56:07,860 --> 00:56:09,410
And if it's not 0,
you have a drift,
678
00:56:09,410 --> 00:56:12,132
so it's not a martingale.
679
00:56:12,132 --> 00:56:13,590
That gives you some
classification.
680
00:56:13,590 --> 00:56:15,390
Now, if you look at a
differential equation
681
00:56:15,390 --> 00:56:17,310
of this stochastic--
this is called
682
00:56:17,310 --> 00:56:19,791
a stochastic differential
equation-- if you know
683
00:56:19,791 --> 00:56:22,290
stochastic process, if you look
at a stochastic differential
684
00:56:22,290 --> 00:56:26,440
equation, if it doesn't have a
drift term, it's a martingale.
685
00:56:26,440 --> 00:56:29,510
If it has a drift term,
it's not a martingale.
686
00:56:29,510 --> 00:56:32,310
That'll be really useful
later, so try to remember it.
687
00:56:32,310 --> 00:56:34,290
The whole point is
when you write down
688
00:56:34,290 --> 00:56:36,990
a stochastic process in
terms of something times dt,
689
00:56:36,990 --> 00:56:40,200
something times dB_t,
really this term
690
00:56:40,200 --> 00:56:45,640
contributes towards the
tendency, the slope of whatever
691
00:56:45,640 --> 00:56:47,320
is going to happen
in the future.
692
00:56:47,320 --> 00:56:50,595
And this is like
the variance term.
693
00:56:50,595 --> 00:56:54,430
It adds some variance to
your stochastic process.
694
00:56:54,430 --> 00:56:58,990
But still, it doesn't add
or subtract value over time,
695
00:56:58,990 --> 00:57:03,895
it fairly adds variation.
696
00:57:06,540 --> 00:57:07,150
Remember that.
697
00:57:07,150 --> 00:57:09,890
That's very important fact.
698
00:57:09,890 --> 00:57:11,900
You're going to use it a lot.
699
00:57:11,900 --> 00:57:14,430
For example, you're going to
use it for pricing theory.
700
00:57:14,430 --> 00:57:18,870
In pricing theory, you come up
with this stochastic process
701
00:57:18,870 --> 00:57:20,150
or some strategy.
702
00:57:20,150 --> 00:57:22,130
You look at its value.
703
00:57:22,130 --> 00:57:27,280
Let's say X_t is your value
of your portfolio over time.
704
00:57:27,280 --> 00:57:34,940
If that portfolio has-- then you
match it with your financial--
705
00:57:34,940 --> 00:57:36,830
let me go over it slowly again.
706
00:57:36,830 --> 00:57:44,780
First you have a financial
derivative, like option
707
00:57:44,780 --> 00:57:47,632
of a stock.
708
00:57:47,632 --> 00:57:49,215
Then you have your
portfolio strategy.
709
00:57:55,630 --> 00:57:57,720
Assume that you have
some strategy that,
710
00:57:57,720 --> 00:57:59,940
at the expiration
time, gives you
711
00:57:59,940 --> 00:58:01,310
the exact value of the option.
712
00:58:03,820 --> 00:58:05,820
Now you look at the
difference between these two
713
00:58:05,820 --> 00:58:06,695
stochastic processes.
714
00:58:10,940 --> 00:58:14,880
Basically what the thing is,
when your variance goes to 0,
715
00:58:14,880 --> 00:58:19,310
your drift also has to go to 0.
716
00:58:19,310 --> 00:58:20,920
So when you look
at the difference,
717
00:58:20,920 --> 00:58:24,010
if you can somehow get rid
of this variance term, that
718
00:58:24,010 --> 00:58:26,880
means no matter
what you do, that
719
00:58:26,880 --> 00:58:30,660
will govern the value
of your portfolio.
720
00:58:30,660 --> 00:58:34,084
If it's positive, that means
you can always make money,
721
00:58:34,084 --> 00:58:35,250
because there's no variance.
722
00:58:35,250 --> 00:58:37,280
Without variance,
you make money.
723
00:58:37,280 --> 00:58:41,770
That's called arbitrage,
and you cannot have that.
724
00:58:41,770 --> 00:58:43,980
But I won't go
into further detail
725
00:58:43,980 --> 00:58:46,870
because Vasily will
cover it next time.
726
00:58:46,870 --> 00:58:49,070
But just remember that flavor.
727
00:58:49,070 --> 00:58:51,820
So when you write something down
in a stochastic differential
728
00:58:51,820 --> 00:58:55,790
equation form, that
term is a drift term,
729
00:58:55,790 --> 00:58:57,272
that term is a variance term.
730
00:58:57,272 --> 00:58:59,230
And if you don't have
drift, it's a martingale.
731
00:59:01,876 --> 00:59:03,290
That is very important.
732
00:59:12,290 --> 00:59:12,950
Any questions?
733
00:59:12,950 --> 00:59:16,658
That's kind of the
basics of Ito calculus.
734
00:59:22,520 --> 00:59:26,260
I will give you some
exercises on it,
735
00:59:26,260 --> 00:59:29,520
mostly just basic computation
exercises, so that you'll
736
00:59:29,520 --> 00:59:31,120
get familiar with it.
737
00:59:31,120 --> 00:59:33,210
Try to practice it.
738
00:59:33,210 --> 00:59:38,320
And let me cover one more
thing called Girsanov theorem.
739
00:59:38,320 --> 00:59:40,900
It's related, but
these are really
740
00:59:40,900 --> 00:59:42,750
basics of the Ito
calculus, so if you
741
00:59:42,750 --> 00:59:46,300
have any questions on
this, please ask me
742
00:59:46,300 --> 00:59:48,724
right now before I move
on to the next topic.
743
00:59:56,842 --> 00:59:58,840
The last thing I want
to talk about today.
744
01:00:43,710 --> 01:00:47,007
Here is an underlying question.
745
01:00:47,007 --> 01:00:48,590
Suppose you have two
Brownian motions.
746
01:00:57,050 --> 01:00:58,240
This is without drift.
747
01:01:01,810 --> 01:01:06,905
And you have another B tilde,
Brownian motion with drift.
748
01:01:12,910 --> 01:01:15,530
These are two probability
distributions over paths.
749
01:01:18,290 --> 01:01:21,320
According to B_t, you're
more likely to have
750
01:01:21,320 --> 01:01:25,770
some Brownian motion
that has no drift.
751
01:01:25,770 --> 01:01:28,290
That's a sample path.
752
01:01:28,290 --> 01:01:31,090
According to B tilde,
you have some drift.
753
01:01:34,890 --> 01:01:37,533
Your Brownian motion will--
754
01:01:41,240 --> 01:01:46,190
A typical path will follow this
line and will follow that line.
755
01:01:46,190 --> 01:01:52,920
The question is
this-- can we switch
756
01:01:52,920 --> 01:01:55,490
from this distribution
to this distribution
757
01:01:55,490 --> 01:01:56,561
by a change of measure?
758
01:02:02,250 --> 01:02:12,730
Can we switch between
the two measures
759
01:02:12,730 --> 01:02:23,720
to probability distributions
by a change of measure?
760
01:02:30,990 --> 01:02:34,114
Let me go a little bit
more what it really means.
761
01:02:34,114 --> 01:02:36,280
Assume that you're just
looking at a Brownian motion
762
01:02:36,280 --> 01:02:41,760
from time 0 up to time t,
some fixed time interval.
763
01:02:41,760 --> 01:02:47,610
Then according to B_t, let's
say this is a sample path omega.
764
01:02:47,610 --> 01:02:54,060
You have some probability
of omega-- this is a p.d.f.
765
01:02:54,060 --> 01:03:01,830
given by this Brownian
motion B. And then you
766
01:03:01,830 --> 01:03:06,100
have another p.d.f., P tilde
of omega, which is a p.d.f.
767
01:03:06,100 --> 01:03:11,240
given by B of t.
768
01:03:11,240 --> 01:03:14,990
The question is,
does there exist a Z
769
01:03:14,990 --> 01:03:19,610
depending on omega
such that P of omega
770
01:03:19,610 --> 01:03:23,740
is equal to Z times P tilde?
771
01:03:39,298 --> 01:03:40,589
Do you understand the question?
772
01:03:46,220 --> 01:03:49,420
Clearly, if you just look at
it, they're quite different.
773
01:03:49,420 --> 01:03:52,220
The path that you get
according to the distributions
774
01:03:52,220 --> 01:03:55,230
are quite different.
775
01:03:55,230 --> 01:03:57,740
It's not clear why we
should expect it at all.
776
01:04:02,866 --> 01:04:03,990
You'll see the answer soon.
777
01:04:03,990 --> 01:04:07,260
But let me discuss all this
in a different context.
778
01:04:17,080 --> 01:04:19,880
Just forget about all the
Brownian motion and everything
779
01:04:19,880 --> 01:04:22,410
just for a moment.
780
01:04:22,410 --> 01:04:26,010
In this concept, changing from
one probability distribution
781
01:04:26,010 --> 01:04:29,270
to another distribution,
it's a very important concept
782
01:04:29,270 --> 01:04:32,930
in analysis and probability
just in general, theoretically.
783
01:04:32,930 --> 01:04:39,060
And there's a name for this
Z, for this changing measure.
784
01:04:39,060 --> 01:04:46,380
If Z exists, it's called the
Radon-Nikodym derivative.
785
01:04:50,900 --> 01:04:53,096
Before doing that, let me
talk a little bit more.
786
01:04:59,940 --> 01:05:03,910
Suppose P is a probability
distribution over omega.
787
01:05:09,204 --> 01:05:10,537
It's a probability distribution.
788
01:05:18,660 --> 01:05:21,560
So this is some set, and P
describes the probability
789
01:05:21,560 --> 01:05:25,660
that you have each
element in the set.
790
01:05:25,660 --> 01:05:27,970
And you have another probability
distribution, P tilde.
791
01:05:33,520 --> 01:05:45,760
We define P and P tilde to be
equivalent if the probability
792
01:05:45,760 --> 01:05:50,667
that A is greater than
zero if and only if...
793
01:05:53,800 --> 01:05:54,400
For all...
794
01:05:58,310 --> 01:06:01,150
These probability distributions
describe the probability
795
01:06:01,150 --> 01:06:03,380
of the subsets.
796
01:06:03,380 --> 01:06:05,210
Think about a very simple case.
797
01:06:05,210 --> 01:06:09,620
Sigma is equal to 1, 2, and 3.
798
01:06:09,620 --> 01:06:13,495
P gives 1/3 probability
to 1, 1/3 probability
799
01:06:13,495 --> 01:06:16,970
to 2, 1/3 probability to 3.
800
01:06:16,970 --> 01:06:22,660
P tilde gives 2/3 probability
to 3, 1 over 6 probability
801
01:06:22,660 --> 01:06:26,570
to 2, 1 over 6 probability to 3.
802
01:06:26,570 --> 01:06:29,290
We have two probability
distribution over some space.
803
01:06:32,020 --> 01:06:34,790
They are equivalent
if, whenever you
804
01:06:34,790 --> 01:06:39,210
take a subset of your
background set-- let's say 1, 2.
805
01:06:39,210 --> 01:06:41,560
When A is equal
to 1, 2, according
806
01:06:41,560 --> 01:06:44,810
to probability distribution
P, the probability
807
01:06:44,810 --> 01:06:48,125
you fall into this
set A is equal to 2/3.
808
01:06:50,770 --> 01:06:54,750
According to P
tilde, you have 5/6.
809
01:06:57,460 --> 01:06:58,590
They're not the same.
810
01:06:58,590 --> 01:07:00,460
The probability itself
is not the same,
811
01:07:00,460 --> 01:07:03,220
but this condition is
satisfied when it's 0.
812
01:07:03,220 --> 01:07:04,865
And when it's not 0, it's not 0.
813
01:07:04,865 --> 01:07:07,406
And you can just check that it's
always true, because they're
814
01:07:07,406 --> 01:07:09,360
all positive probabilities.
815
01:07:09,360 --> 01:07:14,270
On the other hand, if
you take instead, say,
816
01:07:14,270 --> 01:07:19,640
1/3 and 0, now you
take your A to be 3.
817
01:07:22,850 --> 01:07:28,610
Then you have 1/3 equal to 0.
818
01:07:28,610 --> 01:07:33,160
This means, according to
probability distribution P,
819
01:07:33,160 --> 01:07:37,320
there is some probability
that you'll get 3.
820
01:07:37,320 --> 01:07:39,990
But according to probability
distribution P tilde,
821
01:07:39,990 --> 01:07:43,970
you don't have any
probability of getting 3.
822
01:07:43,970 --> 01:07:47,620
So they're not
equivalent in this case.
823
01:07:47,620 --> 01:07:49,840
If you think about it,
then it's really clear.
824
01:07:49,840 --> 01:07:52,360
The theorem says-- this is
a very important theorem
825
01:07:52,360 --> 01:07:53,690
in analysis, actually.
826
01:07:57,875 --> 01:08:08,475
The theorem-- there exists a Z
such that P of omega is equal
827
01:08:08,475 --> 01:08:08,975
to...
828
01:08:12,380 --> 01:08:16,638
If and only if P and P
tilde are equivalent.
829
01:08:22,510 --> 01:08:24,750
You can change from
one probability measure
830
01:08:24,750 --> 01:08:27,029
to another probability
measure just
831
01:08:27,029 --> 01:08:32,231
in terms of multiplication, if
and only if they're equivalent.
832
01:08:32,231 --> 01:08:34,189
And you can see that it's
not the case for this
833
01:08:34,189 --> 01:08:35,355
when they're not equivalent.
834
01:08:35,355 --> 01:08:37,740
You can't make a zero
probability to 1/3 probability
835
01:08:37,740 --> 01:08:40,000
by multiplication.
836
01:08:40,000 --> 01:08:44,510
So in the finite world this is
very just intuitive theorem,
837
01:08:44,510 --> 01:08:48,490
but what this is saying is
it's true for all probability
838
01:08:48,490 --> 01:08:50,736
spaces.
839
01:08:50,736 --> 01:08:52,819
And these are called the
Radon-Nikodym derivative.
840
01:09:01,930 --> 01:09:06,990
Our question is, are these two
Brownian motions equivalent?
841
01:09:06,990 --> 01:09:09,915
The paths that this Brownian
motion without drift
842
01:09:09,915 --> 01:09:12,330
takes and the Brownian
motion with drift
843
01:09:12,330 --> 01:09:16,529
takes, are they kind of
the same but just skewed
844
01:09:16,529 --> 01:09:20,562
in distribution, or are they
really fundamentally different?
845
01:09:20,562 --> 01:09:21,538
That's the question.
846
01:09:28,859 --> 01:09:33,479
And what Girsanov's theorem says
is that they are equivalent.
847
01:09:33,479 --> 01:09:36,089
To me, it came as a
little bit non-intuitive.
848
01:09:36,089 --> 01:09:39,880
I would imagine that it's
not equivalent, these two.
849
01:09:39,880 --> 01:09:42,069
These paths have a
very natural tendency.
850
01:09:42,069 --> 01:09:44,870
As it goes to infinity,
these paths and these paths
851
01:09:44,870 --> 01:09:47,779
will really look
a lot different,
852
01:09:47,779 --> 01:09:51,000
because when you go
really, really far,
853
01:09:51,000 --> 01:09:53,939
the paths which have
drift will be just really
854
01:09:53,939 --> 01:09:57,870
close to your line mu of
t, while the paths which
855
01:09:57,870 --> 01:10:00,302
don't have drift will be
really close to the x-axis.
856
01:10:02,900 --> 01:10:06,070
But still, they are equivalent.
857
01:10:06,070 --> 01:10:09,590
You can change from
one to another.
858
01:10:09,590 --> 01:10:13,840
I'll just state that
theorem without proof.
859
01:10:13,840 --> 01:10:17,345
And this will also be
used in pricing theory.
860
01:10:20,930 --> 01:10:23,270
I'm not an expert
enough to tell why,
861
01:10:23,270 --> 01:10:25,140
but basically what
it's saying is,
862
01:10:25,140 --> 01:10:28,580
you switch some
stochastic process
863
01:10:28,580 --> 01:10:31,000
into a stochastic
process without drift,
864
01:10:31,000 --> 01:10:33,610
thus making it
into a martingale.
865
01:10:33,610 --> 01:10:36,180
And martingale has a lot of
meaning in pricing theory,
866
01:10:36,180 --> 01:10:38,310
as you'll see.
867
01:10:38,310 --> 01:10:39,920
This also has application.
868
01:10:39,920 --> 01:10:42,030
That's why I'm trying to
cover it, although it's
869
01:10:42,030 --> 01:10:44,300
quite a technical theorem.
870
01:10:44,300 --> 01:10:46,985
Try to remember at least
a statement and the spirit
871
01:10:46,985 --> 01:10:48,700
of what it means.
872
01:10:48,700 --> 01:10:50,690
It just means these
two are equivalent,
873
01:10:50,690 --> 01:10:52,370
you can change
from one to another
874
01:10:52,370 --> 01:10:53,830
by a multiplicative function.
875
01:11:08,267 --> 01:11:09,850
Let me just state
it in a simple form.
876
01:11:12,615 --> 01:11:14,740
GUEST SPEAKER: If I could
just interject a comment.
877
01:11:14,740 --> 01:11:15,406
PROFESSOR: Sure.
878
01:11:15,406 --> 01:11:19,830
GUEST SPEAKER: With
these changes of measure,
879
01:11:19,830 --> 01:11:24,620
it turns out that all of these
theories with continuous time
880
01:11:24,620 --> 01:11:27,530
processes should have an
interpretation if you've
881
01:11:27,530 --> 01:11:30,720
discretized time,
and should consider
882
01:11:30,720 --> 01:11:34,140
sort of a finer and finer
discretization of the process.
883
01:11:34,140 --> 01:11:40,680
And with this change of measure,
if you consider problems
884
01:11:40,680 --> 01:11:45,840
in discrete stochastic
processes like random walks,
885
01:11:45,840 --> 01:11:52,160
basically how-- say if you're
gambling against a casino
886
01:11:52,160 --> 01:11:54,510
or against another
player, and you
887
01:11:54,510 --> 01:11:58,200
look at how your winnings
evolve as a random walk,
888
01:11:58,200 --> 01:11:59,780
depending on your
odds, your odds
889
01:11:59,780 --> 01:12:03,340
could be that you
will tend to lose.
890
01:12:03,340 --> 01:12:06,740
So there's basically
a drift in your wealth
891
01:12:06,740 --> 01:12:08,940
as this random process evolves.
892
01:12:08,940 --> 01:12:15,820
You can transform that process,
basically by taking out
893
01:12:15,820 --> 01:12:19,560
your expected losses,
to a process which
894
01:12:19,560 --> 01:12:22,830
has zero change in expectation.
895
01:12:22,830 --> 01:12:26,840
And so you can convert
these gambling problems
896
01:12:26,840 --> 01:12:32,020
where there's drift to a version
where the process, essentially,
897
01:12:32,020 --> 01:12:34,460
has no drift and
is a martingale.
898
01:12:34,460 --> 01:12:37,240
And the martingale theory in
stochastic process courses
899
01:12:37,240 --> 01:12:38,560
is very, very powerful.
900
01:12:38,560 --> 01:12:41,090
There's martingale
convergence theorems.
901
01:12:41,090 --> 01:12:44,180
So you know that the
limit of the martingale
902
01:12:44,180 --> 01:12:48,750
is-- there's a convergence
of the process,
903
01:12:48,750 --> 01:12:50,530
and that applies here as well.
904
01:12:55,026 --> 01:12:57,234
PROFESSOR: You will see some
surprising applications.
905
01:12:57,234 --> 01:12:59,594
GUEST SPEAKER: Yeah.
906
01:12:59,594 --> 01:13:03,515
PROFESSOR: And try to at
least digest the statement.
907
01:13:08,540 --> 01:13:12,340
When the guest speaker comes
and says by Girsanov theorem,
908
01:13:12,340 --> 01:13:15,660
they actually know what it is.
909
01:13:15,660 --> 01:13:16,410
There's a spirit.
910
01:13:20,190 --> 01:13:21,707
This is a very simple version.
911
01:13:21,707 --> 01:13:23,290
There's a lot of
complicated versions,
912
01:13:23,290 --> 01:13:24,874
but let me just do it.
913
01:13:30,570 --> 01:13:40,025
So P is a probability
distribution over paths
914
01:13:40,025 --> 01:13:41,900
from [0, T] to the infinity.
915
01:13:41,900 --> 01:13:48,860
What this means is just paths
from that-- stochastic process
916
01:13:48,860 --> 01:13:53,110
defined from time
0 to time T. These
917
01:13:53,110 --> 01:14:10,790
are paths defined by a
Brownian motion with drift mu.
918
01:14:10,790 --> 01:14:14,080
And then P tilde is a
probability distribution
919
01:14:14,080 --> 01:14:16,821
defined by Brownian
motion without drift.
920
01:14:22,600 --> 01:14:27,445
Then P and P tilde
are equivalent.
921
01:14:27,445 --> 01:14:29,320
Not only are they
equivalent, we can actually
922
01:14:29,320 --> 01:14:31,530
compute their
Radon-Nikodym derivative.
923
01:14:36,184 --> 01:14:44,378
And the Radon-Nikodym
derivative Z
924
01:14:44,378 --> 01:14:50,100
which is defined as T of--
which we denote like this
925
01:14:50,100 --> 01:14:51,180
has this nice form.
926
01:15:05,920 --> 01:15:08,150
That's a nice closed form.
927
01:15:08,150 --> 01:15:13,120
Let me just tell you a
few implications of this.
928
01:15:31,490 --> 01:15:35,790
Now, assume you have
some, let's say, value
929
01:15:35,790 --> 01:15:37,430
of your portfolio over time.
930
01:15:37,430 --> 01:15:40,230
That's the stochastic process.
931
01:15:40,230 --> 01:15:44,090
And you measure it according to
this probability distribution.
932
01:15:44,090 --> 01:15:45,930
Let's say it depends
on some stock price
933
01:15:45,930 --> 01:15:47,970
as the stock price is
modeled using a Brownian
934
01:15:47,970 --> 01:15:51,140
motion with drift.
935
01:15:51,140 --> 01:15:53,500
What this is saying
is, now, instead
936
01:15:53,500 --> 01:15:57,920
of computing this expectation
in your probability space--
937
01:15:57,920 --> 01:16:03,140
so this is defined over
the probability space P,
938
01:16:03,140 --> 01:16:06,510
our sigma-- (omega, P)
defined by this probability
939
01:16:06,510 --> 01:16:07,610
distribution.
940
01:16:07,610 --> 01:16:25,730
You can instead
compute it in-- you
941
01:16:25,730 --> 01:16:28,720
can compute as expectation in
a different probability space.
942
01:16:35,080 --> 01:16:38,430
You transform the problems
about Brownian motion with drift
943
01:16:38,430 --> 01:16:41,420
into a problem about Brownian
motion without a drift.
944
01:16:41,420 --> 01:16:43,170
And the reason I have
Z tilde instead of Z
945
01:16:43,170 --> 01:16:45,230
here is because I flipped.
946
01:16:45,230 --> 01:16:54,480
What you really should have is Z
tilde here as expectation of Z.
947
01:16:54,480 --> 01:16:59,380
If you want to use this Z.
948
01:16:59,380 --> 01:17:03,350
I don't expect you to really
be able to do computations
949
01:17:03,350 --> 01:17:06,650
and do that just by looking
at this theorem once.
950
01:17:06,650 --> 01:17:09,384
Just really trying to
digest what it means
951
01:17:09,384 --> 01:17:13,050
and understand the flavor of
it, that you can transform
952
01:17:13,050 --> 01:17:14,650
problems in one
probability space
953
01:17:14,650 --> 01:17:16,990
to another probability space.
954
01:17:16,990 --> 01:17:19,460
And you can actually do that
when the two distributions are
955
01:17:19,460 --> 01:17:22,700
defined by Brownian motions
when one has drift and one
956
01:17:22,700 --> 01:17:25,000
doesn't have a drift.
957
01:17:25,000 --> 01:17:27,700
How we're going
to use it is we're
958
01:17:27,700 --> 01:17:29,802
going to transform a
non-martingale process
959
01:17:29,802 --> 01:17:30,885
into a martingale process.
960
01:17:35,314 --> 01:17:36,730
When you change
into martingale it
961
01:17:36,730 --> 01:17:39,662
has very good physical
meanings to it.
962
01:17:43,450 --> 01:17:44,680
That's it for today.
963
01:17:44,680 --> 01:17:48,030
And you only have one more
math lecture remaining
964
01:17:48,030 --> 01:17:51,580
and maybe one or two
homeworks but if you have two,
965
01:17:51,580 --> 01:17:54,950
the second one
won't be that long.
966
01:17:54,950 --> 01:17:57,340
And you'll have a lot of
guest lectures, exciting guest
967
01:17:57,340 --> 01:18:00,990
lectures, so try
not to miss them.